As we shall see bellow, one of the form of TDTF consists of a hydrodynamical-like system of equations and it is natural to consider a derivation from the quantum Wigner equation within its Vlasov semi-classical approximation [1]}. Nonetheless, we shall derive it from Time Dependent DFT principles using the Lagrangian formalism just to have a better picture of the approximations involved and to be aware of the connections with TF theory.
